Preheat your oven to 400°F (200°C). Lightly grease a 9x13 inch baking dish.

In a large skillet or Dutch oven, melt 3 tbsp of butter over medium heat. Add the diced onion, carrots, and celery. Cook, stirring occasionally, until the vegetables have softened, about 6-8 minutes.

Add the minced garlic and cook for 1 minute more until fragrant.

Sprinkle the 1/3 cup of all-purpose flour over the vegetables and stir constantly for 1 minute to cook out the raw flour taste. This will form a thick paste (a roux).

Slowly and gradually whisk in the chicken broth until smooth, then whisk in the whole milk. Bring the mixture to a simmer, stirring frequently, and cook for 3-5 minutes, until the sauce has thickened enough to coat the back of a spoon.

Remove the skillet from the heat. Stir in the shredded chicken, frozen peas, thyme, 1/2 tsp salt, and 1/4 tsp pepper. Taste and adjust seasoning if needed. Pour the filling evenly into the prepared baking dish.

In a large mixing bowl, whisk together 2 cups of flour, baking powder, 1/2 tsp salt, and sugar.

Add the cold, cubed butter to the flour mixture. Using a pastry blender or your fingertips, cut the butter into the flour until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.

Stir in the shredded cheddar cheese. Pour in the buttermilk and stir with a fork just until a shaggy, soft dough forms. Be careful not to overmix.

Drop large spoonfuls of the biscuit dough evenly over the top of the chicken filling. You should have about 8-10 biscuits.

Gently brush the tops of the biscuits with the 2 tbsp of melted butter.

Bake for 25-30 minutes, or until the filling is hot and bubbly and the biscuit topping is golden brown and cooked through.

Let the chicken cobbler casserole rest for 5-10 minutes before serving. This allows the creamy sauce to set slightly.